The proposed research bridges mathematics and biology by developing and applying rigidity theory, a branch of mathematics previously applied to engineering and material science, to characterize rigid regions and flexibility in proteins. These rigid regions can have stressed parts, with more constraints than are required for stability, as well as regions that are just rigid. Flexible linkages between the rigid regions allow functionally important motions to occur. The rigidity of frameworks has been studied as part of mathematics for over a century. It is proposed to use this theory to develop an accurate and natural representation of proteins, coupled with modern computational techniques for simulating and visualizing protein motion. This novel collaboration, involving a mathematician, a physicist, and a biochemist, will allow fundamental mathematical questions and important biological applications to be addressed. There will be four focus areas involving mathematics, physics and biology. Representing the interactions in proteins accurately to predict their flexible and rigid regions, which involves applying the molecular frameworks conjecture to three-dimensional protein bond networks, and developing tensegrity theory to represent proteins even more realistically. This constraint-network approach will be used to identify the specific interactions within proteins, and across evolution, that especially stabilize or destabilize the structure. Flexibility predictions, Monte Carlo methods, and polygonal chain unfolding theory will be incorporated to simulate the processes of protein folding and unfolding and to examine the phase transition behavior. These dynamic approaches will also be used to probe how proteins and their molecular partners (ligands) flex in order to optimize binding together. These applications of rigidity theory will be used in conjunction with realistic potentials to visualize the movements accessible to proteins and their complexes with ligands. The results of this research, in the form of new algorithms and three-dimensional maps and animations of protein flexibility, will be published and made available on the web. Thus, the mathematical developments will focus in two areas. One is advancing rigidity theory, through rigorously proving and applying the molecular framework conjecture and extending the realism of molecular frameworks through tensegrity frameworks (which incorporates inequalities as constraints for contraction and expansion). Secondly, three-dimensional polygonal chain unfolding, an emerging area of mathematics and computational geometry, will be explored under a given set of constraints. In biology, three challenging areas are being addressed: simulating protein unfolding pathways, including the unfolding phase transition; identifying key stabilizing interactions within protein frameworks; and modeling functionally important flexibility upon protein-ligand binding. Therefore, this research tackles several major challenges in mathematics and biology, while coupling the mathematical advances to biological applications.